It is now well-known that the wake transition regime for a circular cylinder involves two modes of small-scale three-dimensional instability (modes “A” and “B”; Williamson, 1988), depending on the regime of Reynolds number (Re), although almost no understanding of the physical origins of these instabilities, or indeed their effects on near wake formation, have hitherto been made clear. There is now some strong interest in this problem, coming not only from experiment, but also from Direct Numerical Simulation, where, in some cases, these modes A and B have been found clearly (Thompson & Hourigan, 1996; Zhang et al., 1995; Henderson, 1995; Mittal & Balachandar, 1996). Much of the recent surge of activity concerning the wake transition and development of turbulence in wakes has been addressed comprehensively in a review paper. Williamson (1996a).

Three-dimensional wake transition

The Kelvin Helmholtz roll up of three dimensional, temporally evolving, plane mixing layers were simulated numerically. All simulations were begun from a few low wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity profile. The spanwise disturbance wavelength was taken to be less than or equal to the streamwise wavelength associated with the Kelvin Helmholtz roll up. A standard set of clean structures develop in most of the simulations. The spanwise vorticity rolls up into a corrugated spanwise roller, with vortex stretching creating strong spanwise vorticity in a cup shaped region at the vends of the roller. Predominantly streamwise rib vortices develop in the braid region between the rollers. For sufficiently strong initial three dimensional disturbances, these ribs collapse into compact axisymmetric vortices. The rib vortex lines connect to neighboring ribs and are kinked in the opposite direction of the roller vortex lines. Because of this, these two sets of vortex lines remain distinct. For certain initial conditions, persistent ribs do not develop. In such cases the development of significant three dimensionality is delayed. When the initial three dimensional disturbance energy is about equal to, or less than, the two dimensional fundamental disturbance energy, the evolution of the three dimensional disturbance is nearly linear (with respect to the mean and the two dimensional disturbances), at least until the first Kelvin Helmholtz roll up is completed.

The three-dimensional evolution of a plane mixing layer - The Kelvin-Helmholtz rollup

The three‐dimensional (3‐D) instability of a two‐dimensional (2‐D) flow with elliptical streamlines has been proposed as a generic mechanism for the breakdown of many 2‐D flows. A physical interpretation for the mechanism is presented together with an analytical treatment of the problem. It is shown that the stability of an elliptical flow is governed by an Ince equation. An analytical representation for a localized solution is given and establishes a direct link with previous computations and experiments.

On the three-dimensional instability of strained vortices

In this paper, we investigate the three-dimensional instability of a counter-rotating vortex pair to short waves, which are of the order of the vortex core size, and less than the inter-vortex spacing. Our experiments involve detailed visualizations and velocimetry to reveal the spatial structure of the instability for a vortex pair, which is generated underwater by two rotating plates. We discover, in this work, a symmetry-breaking phase relationship between the two vortices, which we show to be consistent with a kinematic matching condition for the disturbances evolving on each vortex. In this sense, the instabilities in each vortex evolve in a coupled, or ‘cooperative’, manner. Further results demonstrate that this instability is a manifestation of an elliptic instability of the vortex cores, which is here identified clearly for the first time in a real open flow. We establish a relationship between elliptic instability and other theoretical instability studies involving Kelvin modes. In particular, we note that the perturbation shape near the vortex centres is unaffected by the finite size of the cores. We find that the long-term evolution of the flow involves the inception of secondary transverse vortex pairs, which develop near the leading stagnation point of the pair. The interaction of these short-wavelength structures with the long-wavelength Crow instability is studied, and we observe significant modifications in the longevity of large vortical structures.

Cooperative elliptic instability of a vortex pair.

Three‐dimensional flows of an inviscid incompressible fluid and an inviscid subsonic compressible gas are considered and it is demonstrated how the WKB method can be used for investigating their stability. The evolution of rapidly oscillating initial data is considered and it is shown that in both cases the corresponding flows are unstable if the transport equations associated with the wave which is advected by the flow have unbounded solutions. Analyzing the corresponding transport equations, a number of classical stability conditions are rederived and some new ones are obtained. In particular, it is demonstrated that steady flows of an incompressible fluid and an inviscid subsonic compressible gas are unstable if they have points of stagnation.

Local stability conditions in fluid dynamics

The recent theory describing 3‐D exact solutions of the Navier–Stokes equations is applied to the problem of stability of 2‐D viscous flow with elliptical streamlines. An intrinsically inviscid instability mechanism persists in all such flows provided the length scale of the disturbance is sufficiently large. Evidence is presented that this mechanism may be responsible for 3‐D instabilities in high Reynolds number flows whose vortex structures can be locally described by elliptical streamlines.

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M. J. Landman

Papers

The three‐dimensional instability of strained vortices in a viscous fluid